The second conjugate space of a Banach algebra as an algebra

Civin, Paul; Yood, Bertram

doi:10.2140/pjm.1961.11.847

Cited by 186 publications

(161 citation statements)

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“…When (1) is taken as the product on M (i.e., X = ,s#), then (4) becomes the Arens product on sl nt * [5] (which coincides with the operator product when sd is as in Lemma 1). Furthermore, (3) defines a right Banach ,s#* # -module structure on X** that gives an antihomomorphism m # :sG.…”

Section: Associated With the Module Multiplicationmentioning

confidence: 98%

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A noncommutative theory of Bade functionals

Hadwin

Orhon

1991

Glasgow Math. J.

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24]). Via the Stone representation space of the Boolean algebra, the theory can be studied through Banach modules over C(K), where K is a compact Hausdorff space. One of the key concepts in the theory is the notion of Bade functionals. If X is a Banach C(^)-module and x e X, then a Bade functional of x with respect to C{K) is a continuous linear functional a-on A" such that, for each a in C(K) with a > 0, we haveIt is clear that the definition of Bade functionals makes sense when C(K) is replaced by an arbitrary C*-algebra. In this paper we show, using elementary C*-algebraic techniques, that most of the known results on Bade functionals for C(K) carry over to the C*-algebra setting. Moreover, we prove an existence theorem in the C*-algebraic case that is new even in the C(K) setting. This result shows that Bade functionals always exist in many important cases, e.g. when the C*-algebra is separable or when the Banach space is either separable or the dual of a separable space. In [14], T. A. Gillespie showed that, for a Banach C(K)-modu\e X, Bade functionals always exist if X does not contain a copy of c 0 , and he asked if the converse is true. However Bade functionals always exist when X = c 0) since c 0 is separable.Although the C*-algebraic point of view clarifies many of the results in [11, XVII.3, XVIII.3], this paper is written for a readership that is not assumed to be expert in C*-algebras. We therefore provide a brief account of the properties of C*-algebras and of Arens extensions that we need.Throughout, si will denote a C*-algebra with 1. The Gelfand-Naimark theorem says that we can assume that si is a C*-subalgebra of the algebra L(H) of all (bounded linear) operators on a Hilbert space H. We will choose H to have the additional properties listed in the following lemma. The interested reader can consult [22] for details. If AT is a Banach space, we follow the notation of [15] and let X** denote the normed dual of X. We will use si' to denote the commutant of si, and we use * to denote the involution in a C*-algebra. However, we still use w* to denote the weak-star topology.

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Section: Associated With the Module Multiplicationmentioning

confidence: 98%

“…The following lemma, which is based on the polar decomposition of linear functional on a C*-algebra (see [22]), is our main tool in proving the existence of Bade functionals. LEMMA 5. If x eX and aeX**, then there is a partial isometry B in si**** such that, for each A in si with A^O,…”

Section: Associated With the Module Multiplicationmentioning

confidence: 99%

A noncommutative theory of Bade functionals

Hadwin

Orhon

1991

Glasgow Math. J.

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“…Comme F -^ n est continue, l'application .^(F) -> n introduite dans la démontration du théorème V est continue pour les topologies o-(^ld, (3) et cr( n, n'). L'image de la boule unité de ^ll^(F) est l'enveloppe convexe équilibrée de F. D'après le théorème de Krein ( [10], § 8-13), cet ensemble est relativement faiblement compact et l'application Jl^(r) -> n est faiblement compacte. Sa transposée n' -> (°(r) c (Jt^(r))' est aussi faiblement compacte.…”

Section: Doncunclassified

Espaces $Lp$ relatifs à une famille de mesures

Bertrandias¹

1971

Annales De L’institut Fourier

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“…Civin and Yood [4] shows that the Stone-Čech compactification of a discrete semigroup could be given a semigroup structure, which need not be commutative on and is continuous in the left-hand variable; (that is for fixed ∈ , the map → : → is continuous).Indeed the operation on extends uniquely to , so that contained in it's topological center [5]. Pym [7] introduced the concept of an oid (see Section 2 for precise definition).…”

Section: Introductionmentioning

confidence: 99%

Compact Topological Semigroups Associated With Oids

AMINPOUR¹,

Seilani²

2014

J. Math. Computer Sci.

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The known theory for a discrete oid shows that how to find a subset ∞ of which is a compact right topological semigroup (see section 2 for details).In this paper we try to find an analogue of almost periodic functions for oids. We discover, new compact semigroups by using a certain subspace of functions ∞ ( ) of ( ) for an oid for which is continuous on ∞ × ( ∪ ∞ ∪ ∞ ),where( ∪ ∞ ∪ ∞ ) is a suitable subspace of for a wide range.Mathematical Society Classification:2010, 54D35.

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The second conjugate space of a Banach algebra as an algebra

Cited by 186 publications

References 6 publications

A noncommutative theory of Bade functionals

A noncommutative theory of Bade functionals

Espaces $Lp$ relatifs à une famille de mesures

Compact Topological Semigroups Associated With Oids

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